Computability

An arbitrary set M is:
decidable
if an algorithm exists that decides for every x whether x∈M or x∉M. In other words, if the property of its consistency or inconsistency is determinable.
M is decidable ⇔ ΧM is computable.
Where ΧM(x) = {1 if x∈M, 0 if x∉M} is the characteristic function of M.
undecidable
if M is not decidable.
semi-decidable
if an algorithm exists that decides for every x∈M that x∈M. For those x∉M, the algorithm does not even need to terminate. Another terminus for this property is recursively enumerable.
M is semi-decidable ⇔ Χ'M is computable.
Where ΧM'(x) = {1 if x∈M, ⊥ if x∉M} is the semi-characteristic function of M.